
where \(^*\)  stands for the complex conjugate, and \(\mathcal {C}_f\)  is a smooth function that decays rapidly away from the origin. To fully determine the forcing \(f\in , we furthermore take \) E[ f(t,x)f(t',x')]=0\(, which implies that its real and imaginary parts are chosen independently. For analytical and numerical purposes, we will for instance consider \) Cf(x)=(-x22L2)\(, where the large length scale \) L\( will eventually coincide with the correlation length scale of velocity \) u\(, and is known in turbulence phenomenology as the integral length scale.\) Several operators and parameters enter in the dynamics of the velocity field \(u(t,x)\)  (Eq. ). Let us begin with the operator \(\mathcal {L}\)  that, as we will see, is responsible for the transfer of energy from the large scale \(L\)  towards smaller ones. The crucial step, made in Ref. [1]} in a discrete setup related to the dynamics of a shell model and in Ref. [2]} in a continuous one insightfully related to the propagation of waves in rotational and stratified flows, lies in demonstrating that such a transfer of energy through scales can be done in a linear fashion. In our continuous set up, we thus consider the linear operator
\(\mathcal {L}u(t,x) \equiv 2i\pi cxu(t,x),\) 
